Theory of Quantum Information with Memory

8 Relative and conditional entropies

8.1 Quantum relative entropy

As a motivation for developing quantum relative entropy, we first recall absolute continuity of measures (see Halmos [57] and Rudin [133]) and Kullback–Leibier divergence (see Kullback–Leibier [101]) below.

Definition 8.1.1 (Absolute continuity and Radon–Nikodym derivative).

Let P and Q be probability measures on a (Borel) measurable space $\mathrm{(}\mathbb{X}\mathrm{,}\mathcal{B}\mathrm{(}\mathbb{X}\mathrm{)}\mathrm{)}$. The measure P is said to be absolutely continuous with respect to measure Q if, for all $\mathit{A}\in \mathcal{B}\mathrm{(}\mathbb{X}\mathrm{)}$, $\mathit{Q}\mathrm{(}\mathit{A}\mathrm{)}=0$, implies $\mathit{P}\mathrm{(}\mathit{A}\mathrm{)}=0$. In this case, there exists a measurable function called the Radon–Nikodym derivative and denoted symbolically by $\frac{\mathit{d}\mathit{P}}{\mathit{d}\mathit{Q}}\mathrm{(}\mathit{x}\mathrm{)}\mathrm{,}\mathit{x}\in \mathbb{X}$, such that

(8.1)$\begin{array}{c}\mathit{P}\mathrm{(}\mathit{A}\mathrm{)}=\underset{\mathit{A}}{\int }{\displaystyle \frac{\mathit{d}\mathit{P}}{\mathit{d}\mathit{Q}}}\mathrm{(}\mathit{x}\mathrm{)}\mathit{Q}\mathrm{(}\mathit{d}\mathit{x}\mathrm{)}\mathrm{,}\forall \mathit{A}\in \mathcal{B}\mathrm{(}\mathbb{X}\mathrm{)}.\end{array}$

Definition 8.1.2 (Kullback–Leibier ...